A First Course in Linear Algebra, 2017a

Chapter 1
Systems of Equations
1.1 Systems of Equations, Geometry
Outcomes
A. Relate the types of solution sets of a system of two (three) variables to the intersections of
lines in a plane (the intersection of planes in three space)
As you may remember, linear equations like 2x +3y = 6 can be graphed as straight lines in the coordinate
plane. We say that this equation is in two variables, in this case x and y. Suppose you have two such
equations, each of which can be graphed as a straight line, and consider the resulting graph of two lines.
What would it mean if there exists a point of intersection between the two lines? This point, which lies on
both graphs, gives x and y values for which both equations are true. In other words, this point gives the
ordered pair (x,y) that satisfy both equations. If the point (x,y) is a point of intersection, we say that (x,y)
is a solution to the two equations. In linear algebra, we often are concerned with finding the solution(s)
to a system of equations, if such solutions exist. First, we consider graphical representations of solutions
and later we will consider the algebraic methods for finding solutions.
When looking for the intersection of two lines in a graph, several situations may arise. The following
picture demonstrates the possible situations when considering two equations (two lines in the graph)
involving two variables.
y
y
y
x
One Solution
x
No Solutions
x
Infinitely Many Solutions
In the first diagram, there is a unique point of intersection, which means that there is only one (unique)
solution to the two equations. In the second, there are no points of intersection and no solution. When no
solution exists, this means that the two lines are parallel and they never intersect. The third situation which
can occur, as demonstrated in diagram three, is that the two lines are really the same line. For example,
x + y = 1and2x + 2y = 2 are equations which when graphed yield the same line. In this case there are
infinitely many points which are solutions of these two equations, as every ordered pair which is on the
graph of the line satisfies both equations. When considering linear systems of equations, there are always
three types of solutions possible; exactly one (unique) solution, infinitely many solutions, or no solution.
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